John Wallis, Algebra (London, 1673), Chap. LXVI in Vol. II p.286 of the Latin edition. This speech is taken in its entirety from Wallis' own English translation: Treatise of Algebra (London, 1685). For more on Wallis, see J.F.Scott, The Mathematical Work of John Wallis (New York: Chelsea Publishing Company, 1981), in which Scott writes of Wallis: "... more than any other the precursor of the mighty Newton. ...it seems almost beyond doubt that among his contemporaries at least Wallis was esteemed eminent, ... over a range of subjects which was truly encyclopaedic. Mathematics, mechanics, sound, philology, the phenomena of the tides, even music - in all these his writings give evidence of profound knowledge ..."

This probability argument has a surprisingly respectable pedigree! In addition to the Bernoullis, both Lagrange and (independently) Poisson used it to explain the problem posed by Jean-Charles Callet (1744-99) to Lagrange:

1+x+x2+¼+xm-1

1+x+x2+ ¼+xn-1

=

1-xm

1-xn

= (1-xm)(1+xn+x2n+ ¼)

= 1-xm+xn-xm+n+x2n-¼

so that, putting x=1, with m < n arbitrary positive integers, we have

m

n

=1-1+1-1+1-¼.

Their answer was to look at the sequence of partial sums (for m=3, n=5) of the full series

år=0

arxr = 1+0+0-x3+0+x5+0+0-x8+0+x10+0+ ¼

and apply the Leibniz probability argument to arrive at the sum 3/5. That is, for every five partial sums, taken from the beginning of the series, three of them are equal to 1 and two to 0.

Augustus De Morgan, as late as 1844, affirmed this result as a valid algebraic deduction from the geometric series. Attempting to give foundation for the undeniable usefulness of divergent series in algebra, De Morgan asserted the need to distinguish the algebraic value of a series and the arithmetic sum, which may not exist: "On Divergent Series", Trans. Camb. Phil. Soc. 1844, Vol. 8, Part II, pp. 182-203 (pub. 1849).

Subsequent attempts to answer this question exhibit a great reluctance to dismiss divergent series as illegal immigrants, for they proved themselves useful in some surprising ways. It was widely believed in the eighteenth century that there could be valid equations involving power series which, however, have no numerical interpretation. Put another way, one had to distinguish between the algebraic and the arithmetic meaning of a series. See the remarks under references 12 and 24. For Euler's own attempts to grasp the formal value of a series which is divergent numerically, see Barbeau, E. J., and Leah, P. J., "Euler's 1760 paper on divergent series", Historia Mathematica 3 (1976), pp. 141-160. See also Kline, Mathematical Thought , p.465, where Euler is quoted (in his Institutiones , 1755): "Let us say, therefore, that the sum of any infinite series is the finite expression by the expansion of which the series is generated. In this sense the sum of the infinite series 1-x+x2-x3+¼ will be [1/(1+x)], because the series arises from the expansion of this fraction, whatever number is put in place of x. If this is agreed, the new definition of the word sum coincides with the ordinary meaning when a series converges; and since divergent series have no sum in the proper sense of the word, no inconvenience can arise from this terminology. Finally, by means of this definition, we can preserve the utility of divergent series and defend their use from all objections."

Nicholas Saunderson, The Elements of Algebra (Cambridge, 1741); the entire speech, apart from greeting and valediction, is taken from Vol.1. This work went through five editions (all posthumous) between 1740 and 1792. Saunderson has been described as feisty, forthright, very intelligent, and a diligent teacher. It was said of him at Cambridge that he was a teacher who "had not the use of his eyes but taught others to use theirs." He was married and had two children; he was an accomplished musician and an avid horserider: he would follow a pack of hounds (in spite of his blindness) "not only with ardour but with desperation!" He was passionate, outspoken, more apt to inspire admiration than to make or preserve friends; however, he was an excellent companion and a lively conversationalist, full of wit and vivacity. For more about him, see: J.J.Tattersall, "Nicholas Saunderson: The Blind Lucasian Professor," Historia Mathematica 19 (1992), pp. 356-370.

Leonhard Euler ranks among the very greatest mathematicians, and was certainly the central figure in eighteenth-century mathematics. His name appears all over mathematics: Euler equations, Euler theorems, Euler formulas, Euler polynomials, Euler integrals, Euler constants, Euler lines. He was born near the Swiss city of Basle, the son of a preacher, and studied theology at the University of Basle. He started to work in mathematics through the influence of the Bernoulli family, and published his first paper aged eighteen. He won a prize at the age of nineteen from the French Académie des Sciences, and thus embarked upon his incredible career as the most productive and resourceful - if not the most innovative - mathematician of all time. (Paul Erdös may have succeeded to that title now.) An edition of Euler's collected works, including many textbooks and many hundreds of research papers, runs to over seventy volumes! At the age of twenty-six, Euler was appointed to a post at the St. Petersburg Academy in Russia. He spent eight years there, followed by twenty-five years in the service of Frederick the Great in Berlin, before returning to St. Petersburg under Catherine the Great for the last seventeen years of his life, during which he was totally blind. Euler's response to this disability was: "I'll have fewer distractions!" Assisted by his phenomenal memory powers, he continued to produce books and papers at a prodigious rate, until, on September 7, 1783, he "ceased to calculate and to live" (in the famous words of Condorcet).

Carl Boyer remarks, in A History of Mathematics (New York: Wiley and Sons, 1968): "The exceptional didactic quality of Euler's Algebra is attributed to the fact that it was dictated by the blind author through a relatively untutored domestic".

Leonhard Euler, Elements of Algebra , 1770, extract from the English (4th) edition of 1828. This proved to be a popular textbook, appearing in German and Russian editions at St. Petersburg 1770-1772, in a French edition in 1774, an English edition in 1828, and in various other versions later - including American versions in English, and a twentieth-century German edition.

Laplace's epoch-making Celestial Mechanics appeared in five volumes, 1799-1825, bringing him to the height of his fame as the "Newton of France." As a mathematician he leaned more to the "applied" than most of his contemporaries, regarding mathematics as a set of instruments for the study of the physical world. As a political opportunist, he managed to stay in favour with whoever was in power during the uncertain years of the French Revolution. Here is a recollection by Joseph Fourier, who was a student at the École Normale: "Laplace seems quite young; his voice is quiet and clear and he speaks precisely, though not fluently; his appearance is pleasant, and he dresses very simply; he is of medium height. His teaching of mathematics is in no way remarkable and he covers the material very rapidly."

This is not taken from any published record of his teaching or writing on this topic; he did not publish lecture notes. It is a guess at how he would have taught the topic at the École Normale, based on transcripts of his teaching there, in P. Dedron and J. Itard, Mathematics and Mathematicians (Open University Press, 1978), Vols. 1 and 2.

D'Alembert's article, under the heading "Negative," has been aptly described (by Jacky Sip) as an "articulate but puzzled discussion by the most mathematically sophisticated of the eighteenth-century philosophes ." The first half of this Epilogue speech (up to the words: "isolated negative quantity" and excluding the paragraph referenced as item 24) is taken largely from the two Encyclopédie articles, under "Negative" and "Series."

"The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and paradoxes..."

But Abel remained curious about why such diabolical creatures could be so useful. He goes on in his letter:

"That most of these things are correct in spite of that is extraordinarily surprising. I am trying to find a reason for this; it is an exceedingly interesting question."

Sadly, Abel died of tuberculosis a few years later, and did not find the answer. Augustin-Louis Cauchy followed d'Alembert and Abel in (rather uneasily) banishing divergent series from mathematics. Generally speaking, the French mathematicians followed Cauchy's example; however the English and the Germans would not relinquish their conviction (to be justified much later) that there was something really important mathematically, carried within the algebraic form of such series, quite beyond the question of mere numerical convergence or divergence.

At Cambridge, Robert Woodhouse claimed (in 1803) that the equals sign in the equation:

1

1-r

=1+r+r2+¼

has a "more extended signification" than just numerical equality; and George Peacock (in 1833) regarded this infinite series as representing [1/(1-r)] for all r: "for r=1 we do get ¥ = 1+1+1+¼. For r > 1 we get a negative number on the left and, because the terms on the right continually increase, a quantity more than ¥ on the right [...] The attempt to exclude the use of divergent series in symbolical operations would necessarily impose a limit upon the universality of algebraic formulas and operations which is altogether contrary to the spirit of science [...] It would necessarily lead to a great and embarrassing multiplication of cases: it would deprive almost all algebraic operations of much of their certainty and simplicity."

In Germany, Martin Ohm (1792-1872, the brother of the better-known physicist) systematically distinguished the algebraic and arithmetic meanings, and saw formal, infinite series as special "ideal" elements adjoined to the family of finite algebraic expressions. He wrote: "An infinite series (leaving aside any question of convergence or divergence) is completely suited to represent a given expression if one can be sure of having the correct law of development of the series. Of the value of an infinite series one can speak only if it converges." See Kline, Mathematical Thought , pp. 973-977, and 1096-1121; also Hans Niels Jahnke, "Algebraic Analysis in Germany, 1780-1840: Some Mathematical and Philosophical Issues," Historia Mathematica 20 (1993), pp. 265-284.

"The great torrent of creativity released by Christianity now looked as if it were overflowing its banks, sweeping aside the dikes and dams designed to hold it back .... a Cloud of Knowing gathering ..... the threat was of light, not darkness, - a dazzling, blinding light. This time, God's finger pointed inexorably at Pascal himself." - Malcolm Muggeridge (in The Third Testament).